A violin string vibrates at 300 Hz when unfingered. At what frequency will it vibrate if it is fingered one-third of the way down from the end? (That is, only two-thirds of the string vibrates as a standing wave.) Please help!

The fundamental half-wavelength of the vibrating string will be decreased to 2/3 of the original value. Since the frequency of traveling waves in the string stays the same (wavelength x frequency = constant) , that makes the frquency higher by a factor 3/2. (450 Hz)

To find the frequency at which the string will vibrate when fingered one-third of the way down from the end, we need to understand the concept of harmonic frequencies for a vibrating string.

The harmonic frequencies of a vibrating string are determined by the length of the string that is free to vibrate. When the string is unfingered, the entire length of the string vibrates, resulting in the fundamental frequency.

However, when the string is fingered one-third of the way down, only two-thirds of the string is free to vibrate. This changes the effective length of the string and therefore affects the frequency at which it vibrates.

Let's follow these steps to find the frequency of the fingered string:

Step 1: Find the length of the vibrating portion of the string.
Since only two-thirds of the string is free to vibrate, the length of the vibrating portion will be two-thirds of the original length of the entire string.

Step 2: Calculate the frequency of the fingered string.
The frequency of a vibrating string is inversely proportional to its effective length. This means that if the effective length becomes smaller, the frequency will increase.

The relationship between frequency and length is given by the equation:

f1 / f2 = L2 / L1

where f1 and f2 are the frequencies, and L1 and L2 are the lengths of the string.

Given that the original frequency (unfingered) is 300 Hz, let's represent it as f1. The length of the unfingered string is L1, and the length of the fingered string (two-thirds of the original length) is L2.

Using the formula, we can write the equation as:

300 / f2 = L2 / L1

Step 3: Solve the equation for f2.
Rearranging the equation, we have:

f2 = 300 * (L1 / L2)

Since L2 is two-thirds of L1, we can substitute L2 with (2/3) L1:

f2 = 300 * (L1 / (2/3) L1)

Simplifying further:

f2 = 300 * (3/2)
f2 = 450 Hz

Therefore, the frequency at which the string will vibrate when it is fingered one-third of the way down from the end is 450 Hz.

To find the frequency at which the violin string will vibrate when fingered one-third of the way down from the end, we need to consider the concept of standing waves on a string.

When a string is plucked or bowed, it creates a standing wave pattern with nodes and antinodes. Nodes are points on the string that remain still, while antinodes are points of maximum displacement. The fundamental frequency, or first harmonic, is the lowest frequency at which the string vibrates. For a string vibrating at the fundamental frequency, it has one antinode at each end and no nodes in between.

In this case, since only two-thirds of the string is vibrating, we need to find the length of this shorter vibrating segment. To do this, we subtract one-third of the total length from the total length of the string. Let's denote the total length of the string as L.

Length of vibrating segment = L - (1/3)L = (2/3)L

Next, we need to determine the frequency of vibration. The frequency of a standing wave on a string is inversely proportional to its length, meaning that as the length decreases, the frequency increases. To find the frequency, we can use the equation:
f = v / λ

where:
f is the frequency,
v is the velocity of the wave (which is constant for a given string), and
λ is the wavelength.

In the case of a string vibrating at its fundamental frequency, the wavelength is twice the length of the vibrating segment of the string. Thus, the wavelength can be calculated as:
λ = 2 * (2/3)L = (4/3)L

Since the velocity of the wave remains constant for a given string, we can omit it from the equation.

Now we can find the frequency using the equation:
f = v / λ = v / (4/3)L

However, since the velocity and string length are constant, we can simplify the equation to:
f = constant / L

Therefore, the frequency of vibration when the string is fingered one-third of the way down from the end will be proportional to the reciprocal of the length of the shorter vibrating segment. In this case, the length is (2/3)L, so we can calculate the new frequency:

f' = f * (L / (2/3)L)
f' = (300 Hz) * (L / (2/3)L)

Simplifying further:
f' = (300 Hz) * (3/2)
f' = 450 Hz

Hence, the violin string will vibrate at a frequency of 450 Hz when fingered one-third of the way down from the end.